## Complete convergence in the strong law of large numbers for double sums indexed by a sector with functional boundaries.(Ukrainian, English)Zbl 1050.60034

Teor. Jmovirn. Mat. Stat. 68, 44-48 (2003); translation in Theory Probab. Math. Stat. 68, 49-54 (2004).
Let $$\{X(i,j);\;i\geq1, j\geq1\}$$ be i.i.d. random variables; $$S(m,n)= \sum_{i=1}^{m} \sum_{j=1}^{n}X(i,j)$$; $$f: R\to R$$ be a function such that $$f(x)\geq x$$ for all $$x\geq1$$; $$A=\{(m,n): m\leq n\leq f(m)\}$$; $$\tau_{k}=\text{card}\{(m,n): ij=k, (i,j)\in A\}$$. The sequence of random variables $$\{U(m,n)$$; $$m\geq1$$, $$n\geq1\}$$ is called completely convergent to 0 on the set $$A$$ if $$\sum_{(m,n)\in A}\text{P}(| U(m,n)| \geq\varepsilon)<\infty$$ for all $$\varepsilon>0$$. The main result of this paper is the following: The complete convergence of $$S(m,n)/mn$$ to 0 on the set $$A$$ is equivalent to the conditions $$EX(i,j)=0$$, $$\sum_{k=1}^{\infty}k\tau_{k}\text{P} (| X(i,j)|\geq\delta)<\infty$$ for all $$\delta>0$$.

### MSC:

 60F15 Strong limit theorems 60K05 Renewal theory